Intermittent free turbulence

The distribution of the rate of dissipation of free turbulence, as observed in the ocean and the atmosphere, seldom satisfies the homogeneity assumption of the classic Kolmogorov-Obukhov theory. Analogous “intermittency” (with “clustering” of the “active regions”) is also observed for the energy of various “1/ƒ noises,” for the spatial distribution of rare resources such as metals, and elsewhere. By putting together the existing discussions of the various forms of intermittency, one finds two broad approaches. The first was followed by de Wijs (a geophysicist not known outside of his field), and later by Obukhov, Kolmogorov and Yaglom. The second approach was followed by Mandelbrot (who started in the context of 1/ƒ noises) and by Novikov & Stewart (see M 1967k{N12}). The purpose of the present work is to construct further variants of both broad approaches to intermittency, to unify all the approaches, and to develop them. Of particular interest is that all approaches suggest — in agreement with experience — that any interval containing turbulent energy, when examined more closely, will be found to include inserts that are effectively devoid of turbulence. Mathematically, this may be expressed by saying that turbulence is not carried by intervals, but by “thin” sets with many gaps. Such sets can be characterized as having a “dimension” D less than unity (for a discussion of fractional dimension, see M 1967s). At one extreme, there is no insert, and D=1; at the other extreme, turbulence is concentrated in a single “puff,” and D=0. Another property of intermittency is the exponent Ω that enters in the expression k −Ω, which gives (up to a factor of proportionality) the spectral density of the rate of dissipation ε. Different approaches to turbulence yield different relations between D and Ω. This suggests either that further assumptions are needed to identify the properties of “the” actual turbulence, or that different models of intermittency are to be used in different contexts. To conclude the random variable ε(r), the average dissipation in a sphere of radius r, is never lognormally distributed (which refutes a hasty claim of de Wijs, Kolmogorov, Obukhov and Yaglom). Moments of ε(r) are obtained. In some cases ε(r) is found to have infinite population variance. This implies — again in agreement with experience — that observed values of ε(r) can be very widely scattered.